Random doubly stochastic tridiagonal matrices (Q2841678)

Let NEWLINE\[NEWLINE\mathcal{T}_n=\Bigg\{\left(\begin{smallmatrix} 1-c_1 &c_1 &0 &0 &\hdots &0\\ c_1&1-c_1-c_2&c_2 &0 &\hdots &0 &0\\ 0 &c_2 &1-c_2-c_3&c_3 &\hdots &0 &0\\ \vdots &\vdots &\vdots &\vdots&\ddots &\vdots &\vdots\\ 0 &0 &0 &0 &\hdots &1-c_{n-1}-c_n&c_n\\ 0 &0 &0 &0 &\hdots &c_n &1-c_n\end{smallmatrix}\right) \Big|\;0\leq c_i\in\mathbb R,\;i=1,\dots,n\Bigg\}NEWLINE\]NEWLINE be the compact convex set of tridiagonal doubly stochastic matrices. These arise naturally in probability problems as birth and death chains with a uniform stationary distribution. The authors study ``typical'' matrices \(T\in\mathcal T_n\) chosen uniformly at random in the set \(\mathcal T_n\). A simple algorithm is presented to allow direct sampling from the uniform distribution on \(\mathcal T_n\). Using this algorithm, the elements above the diagonal in \(T\) are shown to form a Markov chain. For large \(n\), the limiting Markov chain is reversible and explicitly diagonalizable with transformed Jacobi polynomials as eigenfunctions. These results are used to study the limiting behavior of such typical birth and death chains, including their eigenvalues and mixing times. The results on a uniform random tridiagonal doubly stochastic matrices are related to the distribution of alternating permutations chosen uniformly at random.